Consider two random variables with cdf F and G and of bounded support. I noted that if $F^{-1}(G(x))/x$ increases, we have that $F^{-1}(G(x))/x\le F^{-1}(G(x))/1=1$. My question is, can the bounded support hypothesis be removed, perhaps with some additional structure on $G$? For example if $G$ is geometric (i.e. $1-q^x$) and the density of $F$ is decreasing. I was not successful in proving it even in such a simple scenario. Thanks for any help.
2026-03-26 07:55:32.1774511732
Speed of divergence at infinity of inverse cdf
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